We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.
Huanhuan Tian. Maoan Han. "Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle." Abstr. Appl. Anal. 2014 (SI66) 1 - 14, 2014. https://doi.org/10.1155/2014/819798